1. Field of the Invention
This invention relates to a method and apparatus for efficiently scaling down data which has been transformed from the real domain.
2. Description of the Related Art
Many types of data, such as radar data, oil well log data and digital image data, can consume a large amount of computer storage space. For example, computerized digital image files can require in excess of 1 MB. Therefore, several formats have been developed which manipulate the data in order to compress it. The discrete cosine transform (DCT) is a known technique for data compression and underlies a number of compression standards.
The mathematical function for a DCT in one dimension is:
                                                                                          s                  ~                                ⁡                                  (                  k                  )                                            =                                                c                  ⁡                                      (                    k                    )                                                  ⁢                                                      ∑                                          n                      =                      0                                                              N                      -                      1                                                        ⁢                                                                          ⁢                                                            s                      ⁡                                              (                        n                        )                                                              ⁢                    cos                    ⁢                                                                                            π                          ⁡                                                      (                                                                                          2                                ⁢                                n                                                            +                              1                                                        )                                                                          ⁢                        k                                                                    2                        ⁢                                                                                                  ⁢                        N                                                                                                                  ,                                                  ⁢                          where              ⁢                                                          ⁢              s              ⁢                                                          ⁢              is              ⁢                                                          ⁢              the              ⁢                                                          ⁢              array              ⁢                                                          ⁢              of              ⁢                                                          ⁢              N              ⁢                                                          ⁢              original              ⁢                                                          ⁢              values                        ,                                          s                ~                            ⁢                                                          ⁢              is              ⁢                                                          ⁢              the                                ⁢                                          ⁢                      array            ⁢                                                  ⁢                          o              ⁢              f                        ⁢                                                  ⁢            N            ⁢                                                  ⁢            transformed            ⁢                                                  ⁢            values            ⁢                                                  ⁢            and            ⁢                                                  ⁢            the            ⁢                                                  ⁢            coefficients            ⁢                                                                      ⁢                                                                    ⁢            c            ⁢                                                  ⁢            are                    ⁢                                          ⁢                                          ⁢                                                    given                ⁢                                                                  ⁢                by                ⁢                                                                  ⁢                                  c                  ⁡                                      (                    k                    )                                                              =                                                                                          1                      N                                                        ⁢                                                                          ⁢                  for                  ⁢                                                                                                    ⁢                                                                                                  ⁢                  k                                =                0                                      ;                                          c                ⁡                                  (                  k                  )                                            =                                                                                          2                      N                                                        ⁢                                                                          ⁢                  for                  ⁢                                                                          ⁢                  k                                >                0                                                    ⁢                                                      (        1        )            Taking for example the manipulation of image data, blocks of data consisting of 8 rows by 8 columns of data samples frequently are operated upon during image resizing processes. Therefore a two-dimensional DCT calculation is necessary. The equation for a two-dimensional DCT where N=8 is:
                                                                                          s                  ~                                ⁡                                  (                                      i                    ,                    j                                    )                                            =                                                c                  ⁡                                      (                                          i                      ,                      j                                        )                                                  ⁢                                                      ∑                                          n                      =                      0                                        7                                    ⁢                                                                          ⁢                                                            ∑                                              m                        =                        0                                            7                                        ⁢                                                                                  ⁢                                                                  s                        ⁡                                                  (                                                      m                            ,                            n                                                    )                                                                    ⁢                      cos                      ⁢                                                                                          ⁢                                                                                                    π                            ⁡                                                          (                                                                                                2                                  ⁢                                  m                                                                +                                1                                                            )                                                                                ⁢                          i                                                16                                            ⁢                      cos                      ⁢                                                                                          ⁢                                                                                                    π                            ⁡                                                          (                                                                                                2                                  ⁢                                  m                                                                +                                1                                                            )                                                                                ⁢                          j                                                16                                                                                                                  ,                                                  ⁢                                          where                ⁢                                                                  ⁢                s                ⁢                                                                  ⁢                is                ⁢                                                                  ⁢                an                ⁢                                                                  ⁢                8                ×                8                ⁢                                                                  ⁢                matrix                ⁢                                                                  ⁢                of                ⁢                                                                  ⁢                64                ⁢                                                                  ⁢                values                            ;                                                s                  ~                                ⁢                                                                  ⁢                is                ⁢                                                                  ⁢                an                ⁢                                                                  ⁢                8                ×                8                ⁢                                                                  ⁢                matrix                                              ⁢                                          ⁢                      of            ⁢                                                  ⁢            64            ⁢                                                  ⁢            coefficients            ⁢                                                  ⁢            and            ⁢                                                  ⁢            the            ⁢                                                  ⁢            constants            ⁢                                                  ⁢                          c              ⁡                              (                                  i                  ,                  j                                )                                      ⁢                                                  ⁢            are            ⁢                                                  ⁢            given            ⁢                                                  ⁢            by                    ⁢                                          ⁢                                                    c                ⁡                                  (                                      i                    ,                    j                                    )                                            =                              1                8                                      ,                                                            when                  ⁢                                                                          ⁢                  i                                =                                                      0                    ⁢                                                                                  ⁢                    and                    ⁢                                                                                  ⁢                    j                                    =                  0                                            ;                                                c                  ⁡                                      (                                          i                      ,                      j                                        )                                                  =                                                                            1                                              4                        ⁢                                                  2                                                                                      ⁢                                                                                  ⁢                    if                    ⁢                                                                                  ⁢                    i                                    =                                      0                    ⁢                                                                                  ⁢                    and                                                                                ⁢                                          ⁢                                                                      j                  >                                      0                    ⁢                                                                                  ⁢                    or                    ⁢                                                                                  ⁢                    i                                    >                                      0                    ⁢                                                                                  ⁢                    and                    ⁢                                                                                  ⁢                    j                                                  =                0                            ;                                                c                  ⁡                                      (                                          i                      ,                      j                                        )                                                  =                                                      1                    4                                    ⁢                                                                          ⁢                  when                  ⁢                                                                          ⁢                  i                                                      ,                          j              >              0                                      ⁢                                                      (        2        )            Because data is taken from the “real” or spatial image domain and transformed into the DCT domain by equations (1) and (2), these DCT operations are referred to as forward Discrete Cosine Transforms (FDCT), or forward transform operations.
As previously mentioned, the DCT is an image compression technique which underlies a number of compression standards. These include the well-known Joint Photographic Experts Group (JPEG) and the Moving Picture Experts Group (MPEG) standards. Comprehensive references on the JPEG and MPEG standards include JPEG Still Image Data Compression Standard by William B. Pennebaker and Joan L. Mitchell (© 1993 Van Nostrand Reinhold), and MPEG Video Compression Standard by Joan L. Mitchell, William B. Pennebaker, et al (© 1997 Chapman & Hall).
Looking at the JPEG method, for example, there are five basic steps. Again taking the example of the manipulation of image data, the first step is to extract an 8×8 pixel block from the image. The second step is to calculate the FDCT for each block. Third, a quantizer rounds off the DCT coefficients according to the specified image quality. Fourth, the quantized, two-dimensional 8×8 block of DCT coefficients are reordered into a one-dimensional vector according to a zig zag scan order. Fifth, the coefficients are compressed using an entropy encoding scheme such as Huffman coding or arithmetic coding. The final compressed data is then written to the output file.
Returning to the first step, source image samples are grouped into 8×8 data matrices, or blocks. The initial image data is frequently converted from normal RGB color space to a luminance/chrominance color space, such as YUV. YUV is a color space scheme that stores information about an image's luminance (brightness) and chrominance (hue). Because the human eye is more sensitive to luminance than chrominance, more information about an image's chrominance can be discarded as compared to luminance data.
Once an 8×8 data block has been extracted from the original image and is in the desired color scheme, the DCT coefficients are computed. The 8×8 matrix is entered into the DCT algorithm, and transformed into 64 unique, two-dimensional spatial frequencies thereby determining the input block's spectrum.
The ultimate goal of this FDCT step is to represent the image data in a different domain using the cosine functions. This can be advantageous because it is a characteristic of cosine functions that most of the spatial frequencies will disappear for images in which the image data changes slightly as a function of space. The image blocks are transformed into numerous curves of different frequencies. Later, when these curves are put back together through an inverse step, a close approximation to the original block is restored.
After the FDCT step, the 8×8 matrix contains transformed data comprised of 64 DCT coefficients in which the first coefficient, commonly referred to as the DC coefficient, is related to the average of the original 64 values in the block. The other coefficients are commonly referred to as AC coefficients.
Up to this point in the JPEG compression process, little actual image compression has occurred. The 8×8 pixel block has simply been converted into an 8×8 matrix of DCT coefficients. The third step involves preparing the matrix for further compression by quantizing each element in the matrix. The JPEG standard gives two exemplary tables of quantization constants, one for luminance and one for chrominance. These constants were derived from experiments on the human visual system. The 64 values used in the quantization matrix are stored in the JPEG compressed data as part of the header, making dequantization of the coefficients possible. The encoder needs to use the same constants to quantize the DCT coefficients.
Each DCT coefficient is divided by its corresponding constant in the quantization table and rounded off to the nearest integer. The result of quantizing the DCT coefficients is that smaller, unimportant coefficients will disappear and larger coefficients will lose unnecessary precision. As a result of this quantization step, some of the original image quality is lost. However, the actual image data lost is often not visible to the human eye at normal magnification.
Quantizing produces a list of streamlined DCT coefficients that can now be very efficiently compressed using either a Huffman or arithmetic encoding scheme. Thus the final step in the JPEG compression algorithm is to encode the data using an entropy encoding scheme. Before the matrix is encoded, it is arranged in a one-dimensional vector in a zigzag order. The coefficients representing low frequencies are moved to the beginning of the vector and the coefficients representing higher frequencies are placed towards the end of the vector. By placing the higher frequencies (which are more likely to be zeros) at the end of the vector, an end of block code truncates the larger sequence of zeros which permits better overall compression.
Equations (1) and (2) describe the process for performing a FDCT, i.e., taking the data from the real domain into the DCT domain. When it is necessary to reverse this step, i.e., transform the data from the DCT domain to the real domain, a DCT operation known as an Inverse Discrete Cosine Transform (IDCT), or an inverse transform operation, can be performed. For a one-dimensional, inverse transform operation, the IDCT is defined as follows:
                                                                        s                ⁡                                  (                  n                  )                                            =                                                ∑                                      k                    =                    0                                                        N                    -                    1                                                  ⁢                                                                  ⁢                                                      c                    ⁡                                          (                      k                      )                                                        ⁢                                                            s                      ~                                        ⁡                                          (                      k                      )                                                        ⁢                  cos                  ⁢                                                                                    π                        ⁡                                                  (                                                                                    2                              ⁢                              n                                                        +                            1                                                    )                                                                    ⁢                      k                                                              2                      ⁢                                                                                          ⁢                      N                                                                                            ,                                                  ⁢                          where              ⁢                                                          ⁢              s              ⁢                                                          ⁢              is              ⁢                                                          ⁢              the              ⁢                                                          ⁢              array              ⁢                                                          ⁢              of              ⁢                                                          ⁢              N              ⁢                                                          ⁢              original              ⁢                                                          ⁢              values                        ,                                                  ⁢                                          s                ~                            ⁢                                                          ⁢              is              ⁢                                                          ⁢              the                                ⁢                                          ⁢                                          ⁢                      array            ⁢                                                  ⁢            of            ⁢                                                  ⁢            N            ⁢                                                  ⁢            transformed            ⁢                                                  ⁢            values            ⁢                                                  ⁢            and            ⁢                                                  ⁢            the            ⁢                                                  ⁢            coefficients            ⁢                                                  ⁢            c            ⁢                                                  ⁢            are                    ⁢                                          ⁢                                          ⁢                                                    given                ⁢                                                                  ⁢                by                ⁢                                                                  ⁢                                  c                  ⁡                                      (                    k                    )                                                              =                                                                                          1                      N                                                        ⁢                                                                          ⁢                  for                  ⁢                                                                                                    ⁢                                                                                                  ⁢                  k                                =                0                                      ;                                          c                ⁡                                  (                  k                  )                                            =                                                                                          2                      N                                                        ⁢                                                                          ⁢                  for                  ⁢                                                                          ⁢                  k                                >                0                                                    ⁢                                                      (        3        )            For an inverse transform operation in two dimensions where N=8, the IDCT is defined:
                                                        s              ⁡                              (                                  m                  ,                  n                                )                                      =                                          ∑                                  i                  =                  0                                7                            ⁢                                                          ⁢                                                ∑                                      j                    =                    0                                    7                                ⁢                                                                  ⁢                                                      c                    ⁡                                          (                                              i                        ,                        j                                            )                                                        ⁢                                                            s                      ~                                        ⁡                                          (                                              i                        ,                        j                                            )                                                        ⁢                  cos                  ⁢                                                                                    π                        ⁡                                                  (                                                                                    2                              ⁢                                                                                                                          ⁢                              m                                                        +                            1                                                    )                                                                    ⁢                      i                                        16                                    ⁢                  cos                  ⁢                                                                                    π                        ⁡                                                  (                                                                                    2                              ⁢                                                                                                                          ⁢                              n                                                        +                            1                                                    )                                                                    ⁢                      j                                        16                                                                                ,                                          ⁢                      where            ⁢                                                  ⁢            s            ⁢                                                  ⁢            is            ⁢                                                  ⁢            an            ⁢                                                  ⁢            8            ×            8            ⁢                                                  ⁢            matrix            ⁢                                                  ⁢            of            ⁢                                                  ⁢            64            ⁢                                                  ⁢            values                    ,                                          ⁢                                    s              ~                        ⁢                                                  ⁢            is            ⁢                                                  ⁢            an            ⁢                                                  ⁢            8            ×            8            ⁢                                                  ⁢            matrix            ⁢                                                  ⁢            of            ⁢                                                  ⁢            64            ⁢                                                  ⁢            coefficients            ⁢                                                                      ⁢                                                                    ⁢            and            ⁢                                                  ⁢            the            ⁢                                                  ⁢            constants                          ⁢                                  ⁢                                                            c                ⁡                                  (                                      i                    ,                    j                                    )                                            ⁢                                                          ⁢              are              ⁢                                                          ⁢              given              ⁢                                                          ⁢              by              ⁢                                                          ⁢                              c                ⁡                                  (                                      i                    ,                    j                                    )                                                      =                                                            1                  8                                ⁢                                                                                          ⁢                                                                                        ⁢                when                ⁢                                                                  ⁢                i                            =                                                0                  ⁢                                                                          ⁢                  and                  ⁢                                                                          ⁢                  j                                =                0                                              ;                ⁢                                  ⁢                                            c              ⁡                              (                                  i                  ,                  j                                )                                      =                                                            1                                      4                    ⁢                                          2                                                                      ⁢                                                                  ⁢                if                ⁢                                                                  ⁢                i                            =                                                                    0                    ⁢                                                                                  ⁢                    and                    ⁢                                                                                                              ⁢                                                                                                            ⁢                    j                                    >                                      0                    ⁢                                                                                  ⁢                    or                    ⁢                                                                                  ⁢                    i                                    >                                      0                    ⁢                                                                                  ⁢                    and                    ⁢                                                                                  ⁢                    j                                                  =                0                                              ;                ⁢                                  ⁢                                            c              ⁡                              (                                  i                  ,                  j                                )                                      =                                          1                4                            ⁢                                                          ⁢              when              ⁢                                                          ⁢              i                                ,                      j            >            0                                              (        4        )            
As previously stated, digital images are often transmitted and stored in compressed data formats, such as the previously described JPEG standard. In this context, there often arises the need to scale down (i.e., reduce) the dimensions of an image that is provided in a compressed data format in order to achieve a suitable image size.
For example, where an image is to be sent in compressed data format to receivers of different computational and output capabilities, it may be necessary to scale down the size of the image to match the capabilities of each receiver. For example, some printers are designed to receive images which are of a certain size, but the printers must have the capability of scaling down the image size for printing purposes, particularly when the original image was intended for a higher resolution output.
A known method for scaling down an image provided in a compressed data format is illustrated in FIG. 1. First, a determination is made as to the amount of desired image reduction 1/B. (Block 10) If, for example, an image reduction by a factor of 3 is desired (i.e., dividing by 3 or multiplying by ⅓), then 3 adjacent 8×8 blocks of transformed data values (for a total of 64×3 or 192 values) are retrieved from the image. (Block 11) An IDCT is performed on each of the three blocks to transform the pixel or pel data into the real or spatial domain. (Block 12) Once in the real domain, the data is reduced from 3 adjacent 8×8 data blocks into a single 8×8 data block by any one of several, known filtering techniques. (Block 13) Then a FDCT operation is performed on the data of the single 8×8 data block to return the data to the DCT domain. (Block 14) The process is repeated for all remaining data in the input image. (Block 15)
Thus given a portion of an image in a JPEG/DCT compressed data format consisting of four compressed 8×8 blocks of image data, scaling down the image by a factor of two in each dimension using a previously known method requires: (1) entropy decoding the data which is in one-dimensional vector format and placing the data in 8×8 blocks; (2) de-quantizing the data; (3) performing four 8×8 IDCT operations to inverse transform the transformed blocks of image data; (4) additional filtering operations to scale down the blocks of image data into one 8×8 block of scaled image data; (5) an 8×8 FDCT operation to re-transform the block of scaled image data; (6) quantizing the 8×8 block of data; and (7) placing the block of data in one-dimensional vectors and entropy encoding the data for storage or transmission. Given the mathematical complexity of the FDCT and IDCT operations, such a large number of operations is computationally time consuming.
What is needed is an efficient method and apparatus that operates directly upon transformed blocks of image data to convert them into transformed blocks of scaled-down image data.